ccf (daily_bike_data$ Total_Users,daily_bike_data$ Humidity) # no lag
daily_bike_data$ Humidity_lag = dplyr:: lag (daily_bike_data$ Humidity,4 )
# Short Term Prediction Fit
fit1 = lm (Total_Users~ Humidity + Temperature + Day_of_the_Week + Wind_Speed, data = daily_bike_data[1 : 724 ,])
plotts.sample.wge (fit1$ residuals)$ xbar
aic5.wge (fit1$ residuals, type = 'bic' )
---------WORKING... PLEASE WAIT...
Five Smallest Values of bic
p q bic
2 1 13.33609
3 1 13.34050
4 1 13.34482
3 2 13.34737
5 1 13.35261
aic5.wge (fit1$ residuals, type = 'aic' )
---------WORKING... PLEASE WAIT...
Five Smallest Values of aic
p q aic
4 1 13.30683
5 1 13.30828
3 1 13.30884
3 2 13.30937
2 1 13.31076
mlr1 = arima (daily_bike_data$ Total_Users[1 : 724 ], order = c (2 ,0 ,1 ),xreg = cbind (daily_bike_data$ Humidity[1 : 724 ], daily_bike_data$ Temperature[1 : 724 ], daily_bike_data$ Day_of_the_Week[1 : 724 ], daily_bike_data$ Wind_Speed[1 : 724 ]))
plotts.wge (mlr1$ residuals) # looks random
acf (mlr1$ residuals) # none of the acfs out of bounds
ljung.wge (mlr1$ residuals) # greater than 0.05
Obs -0.01990208 0.06690323 -0.04759247 -0.04169753 -0.02500729 0.04925277 0.003767595 0.009962653 0.03051887 0.03138526 -0.02269299 -0.06073938 -0.04083809 0.002403712 0.05638118 -0.05626742 0.02872016 -0.06706423 0.0002921509 0.009292555 -0.01319121 -0.02447803 -0.02243532 -0.04063492
$test
[1] "Ljung-Box test"
$K
[1] 24
$chi.square
[1] 25.46315
$df
[1] 24
$pval
[1] 0.3809535
ljung.wge (mlr1$ residuals, K = 48 )
Obs -0.01990208 0.06690323 -0.04759247 -0.04169753 -0.02500729 0.04925277 0.003767595 0.009962653 0.03051887 0.03138526 -0.02269299 -0.06073938 -0.04083809 0.002403712 0.05638118 -0.05626742 0.02872016 -0.06706423 0.0002921509 0.009292555 -0.01319121 -0.02447803 -0.02243532 -0.04063492 -0.03423427 0.04182856 0.04846727 0.04919145 0.09929602 -0.04749852 -0.01221122 -0.03073617 -0.03474403 -0.002300149 0.009515319 -0.008485978 0.08264753 -0.07534859 0.001227344 -0.004857766 -0.01268176 0.004604187 0.001022967 -0.03190722 -0.01932466 0.02024435 0.002308987 0.004230412
$test
[1] "Ljung-Box test"
$K
[1] 48
$chi.square
[1] 53.45178
$df
[1] 48
$pval
[1] 0.2728763
mlr_st_pred = predict (mlr1, newxreg = data.frame (daily_bike_data$ Humidity[725 : 731 ], daily_bike_data$ Temperature[725 : 731 ], daily_bike_data$ Day_of_the_Week[725 : 731 ], daily_bike_data$ Wind_Speed[725 : 731 ]), n.ahead = 7 )
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (720 ,731 ), xlab= 'Time' , main = 'Last 7 Day Forecast' , ylab = 'Total Users' )
points (seq (725 ,731 ,1 ),mlr_st_pred$ pred,type = 'l' , col = 'blue' )
mlr_st_ASE= mean ((daily_bike_data$ Total_Users[725 : 731 ] - mlr_st_pred$ pred)^ 2 )
mlr_st_ASE
# Long Term Prediction Fit
fit2 = lm (Total_Users~ Humidity + Temperature + Day_of_the_Week + Wind_Speed, data = daily_bike_data[1 : 671 ,])
plotts.sample.wge (fit2$ residuals)$ xbar
aic5.wge (fit2$ residuals, type = 'bic' )
---------WORKING... PLEASE WAIT...
Five Smallest Values of bic
p q bic
2 1 13.31773
3 1 13.32523
4 1 13.32670
1 2 13.33071
3 2 13.33144
mlr2 = arima (daily_bike_data$ Total_Users[1 : 671 ], order = c (2 ,0 ,1 ),xreg = cbind (daily_bike_data$ Humidity[1 : 671 ], daily_bike_data$ Temperature[1 : 671 ], daily_bike_data$ Day_of_the_Week[1 : 671 ], daily_bike_data$ Wind_Speed[1 : 671 ]))
plotts.wge (mlr2$ residuals) # looks random
acf (mlr2$ residuals[- (1 : 5 )]) # 0/20 acfs out of bounds
ljung.wge (mlr2$ residuals) # greater than 0.05
Obs -0.01227934 0.04886318 -0.06648244 -0.05530172 -0.02617738 0.05545272 0.006892823 0.0262907 0.03822637 0.03183932 -0.01335176 -0.03679148 -0.01804655 0.00694351 0.04204606 -0.03577353 0.01581059 -0.05726426 0.00899322 0.01639788 -0.005884267 -0.008716077 -0.03759271 -0.08217471
$test
[1] "Ljung-Box test"
$K
[1] 24
$chi.square
[1] 23.38034
$df
[1] 24
$pval
[1] 0.4974488
ljung.wge (mlr2$ residuals, K = 48 )
Obs -0.01227934 0.04886318 -0.06648244 -0.05530172 -0.02617738 0.05545272 0.006892823 0.0262907 0.03822637 0.03183932 -0.01335176 -0.03679148 -0.01804655 0.00694351 0.04204606 -0.03577353 0.01581059 -0.05726426 0.00899322 0.01639788 -0.005884267 -0.008716077 -0.03759271 -0.08217471 -0.06159733 0.02692185 0.03004697 0.05562971 0.1012735 -0.0665928 -0.0348977 -0.06596134 -0.03262167 -0.02449251 0.05245193 0.006048363 0.06994222 -0.07952593 0.01032963 -0.003487383 -0.02129563 0.01450759 0.03745105 -0.0301164 -0.006270083 0.004538808 -0.00771238 -0.01776573
$test
[1] "Ljung-Box test"
$K
[1] 48
$chi.square
[1] 57.31135
$df
[1] 48
$pval
[1] 0.1680012
mlr_lt_pred = predict (mlr2, newxreg = data.frame (daily_bike_data$ Humidity[672 : 731 ], daily_bike_data$ Temperature[672 : 731 ], daily_bike_data$ Day_of_the_Week[672 : 731 ], daily_bike_data$ Wind_Speed[672 : 731 ]), n.ahead = 60 )
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (670 ,731 ),xlab= 'Time' , main = 'Last 60 Day Forecast' , ylab = "Total Users" )
points (seq (672 ,731 ,1 ),mlr_lt_pred$ pred,type = 'l' , pch = 15 ,col = 'blue' ,lwd= 1 , lty = 1 )
mlr_lt_ASE= mean ((daily_bike_data$ Total_Users[672 : 731 ] - mlr_lt_pred$ pred)^ 2 )
mlr_lt_ASE
# Forecasting The 4 Exogenous Variables
parzen.wge (daily_bike_data$ Humidity, trunc = 300 )
$freq
[1] 0.001367989 0.002735978 0.004103967 0.005471956 0.006839945 0.008207934
[7] 0.009575923 0.010943912 0.012311902 0.013679891 0.015047880 0.016415869
[13] 0.017783858 0.019151847 0.020519836 0.021887825 0.023255814 0.024623803
[19] 0.025991792 0.027359781 0.028727770 0.030095759 0.031463748 0.032831737
[25] 0.034199726 0.035567715 0.036935705 0.038303694 0.039671683 0.041039672
[31] 0.042407661 0.043775650 0.045143639 0.046511628 0.047879617 0.049247606
[37] 0.050615595 0.051983584 0.053351573 0.054719562 0.056087551 0.057455540
[43] 0.058823529 0.060191518 0.061559508 0.062927497 0.064295486 0.065663475
[49] 0.067031464 0.068399453 0.069767442 0.071135431 0.072503420 0.073871409
[55] 0.075239398 0.076607387 0.077975376 0.079343365 0.080711354 0.082079343
[61] 0.083447332 0.084815321 0.086183311 0.087551300 0.088919289 0.090287278
[67] 0.091655267 0.093023256 0.094391245 0.095759234 0.097127223 0.098495212
[73] 0.099863201 0.101231190 0.102599179 0.103967168 0.105335157 0.106703146
[79] 0.108071135 0.109439124 0.110807114 0.112175103 0.113543092 0.114911081
[85] 0.116279070 0.117647059 0.119015048 0.120383037 0.121751026 0.123119015
[91] 0.124487004 0.125854993 0.127222982 0.128590971 0.129958960 0.131326949
[97] 0.132694938 0.134062927 0.135430917 0.136798906 0.138166895 0.139534884
[103] 0.140902873 0.142270862 0.143638851 0.145006840 0.146374829 0.147742818
[109] 0.149110807 0.150478796 0.151846785 0.153214774 0.154582763 0.155950752
[115] 0.157318741 0.158686731 0.160054720 0.161422709 0.162790698 0.164158687
[121] 0.165526676 0.166894665 0.168262654 0.169630643 0.170998632 0.172366621
[127] 0.173734610 0.175102599 0.176470588 0.177838577 0.179206566 0.180574555
[133] 0.181942544 0.183310534 0.184678523 0.186046512 0.187414501 0.188782490
[139] 0.190150479 0.191518468 0.192886457 0.194254446 0.195622435 0.196990424
[145] 0.198358413 0.199726402 0.201094391 0.202462380 0.203830369 0.205198358
[151] 0.206566347 0.207934337 0.209302326 0.210670315 0.212038304 0.213406293
[157] 0.214774282 0.216142271 0.217510260 0.218878249 0.220246238 0.221614227
[163] 0.222982216 0.224350205 0.225718194 0.227086183 0.228454172 0.229822161
[169] 0.231190150 0.232558140 0.233926129 0.235294118 0.236662107 0.238030096
[175] 0.239398085 0.240766074 0.242134063 0.243502052 0.244870041 0.246238030
[181] 0.247606019 0.248974008 0.250341997 0.251709986 0.253077975 0.254445964
[187] 0.255813953 0.257181943 0.258549932 0.259917921 0.261285910 0.262653899
[193] 0.264021888 0.265389877 0.266757866 0.268125855 0.269493844 0.270861833
[199] 0.272229822 0.273597811 0.274965800 0.276333789 0.277701778 0.279069767
[205] 0.280437756 0.281805746 0.283173735 0.284541724 0.285909713 0.287277702
[211] 0.288645691 0.290013680 0.291381669 0.292749658 0.294117647 0.295485636
[217] 0.296853625 0.298221614 0.299589603 0.300957592 0.302325581 0.303693570
[223] 0.305061560 0.306429549 0.307797538 0.309165527 0.310533516 0.311901505
[229] 0.313269494 0.314637483 0.316005472 0.317373461 0.318741450 0.320109439
[235] 0.321477428 0.322845417 0.324213406 0.325581395 0.326949384 0.328317373
[241] 0.329685363 0.331053352 0.332421341 0.333789330 0.335157319 0.336525308
[247] 0.337893297 0.339261286 0.340629275 0.341997264 0.343365253 0.344733242
[253] 0.346101231 0.347469220 0.348837209 0.350205198 0.351573187 0.352941176
[259] 0.354309166 0.355677155 0.357045144 0.358413133 0.359781122 0.361149111
[265] 0.362517100 0.363885089 0.365253078 0.366621067 0.367989056 0.369357045
[271] 0.370725034 0.372093023 0.373461012 0.374829001 0.376196990 0.377564979
[277] 0.378932969 0.380300958 0.381668947 0.383036936 0.384404925 0.385772914
[283] 0.387140903 0.388508892 0.389876881 0.391244870 0.392612859 0.393980848
[289] 0.395348837 0.396716826 0.398084815 0.399452804 0.400820793 0.402188782
[295] 0.403556772 0.404924761 0.406292750 0.407660739 0.409028728 0.410396717
[301] 0.411764706 0.413132695 0.414500684 0.415868673 0.417236662 0.418604651
[307] 0.419972640 0.421340629 0.422708618 0.424076607 0.425444596 0.426812585
[313] 0.428180575 0.429548564 0.430916553 0.432284542 0.433652531 0.435020520
[319] 0.436388509 0.437756498 0.439124487 0.440492476 0.441860465 0.443228454
[325] 0.444596443 0.445964432 0.447332421 0.448700410 0.450068399 0.451436389
[331] 0.452804378 0.454172367 0.455540356 0.456908345 0.458276334 0.459644323
[337] 0.461012312 0.462380301 0.463748290 0.465116279 0.466484268 0.467852257
[343] 0.469220246 0.470588235 0.471956224 0.473324213 0.474692202 0.476060192
[349] 0.477428181 0.478796170 0.480164159 0.481532148 0.482900137 0.484268126
[355] 0.485636115 0.487004104 0.488372093 0.489740082 0.491108071 0.492476060
[361] 0.493844049 0.495212038 0.496580027 0.497948016 0.499316005
$pzgram
[1] 8.594937115 9.040416537 8.744514441 8.203526707 7.691930774
[6] 6.657928472 4.987353411 3.779066540 3.822037161 4.117778835
[11] 4.080305573 3.900108803 3.650840883 3.286363599 3.166227809
[16] 3.560062953 4.148542595 4.679436500 5.036207992 5.022028359
[21] 4.618223586 4.095094448 3.752299534 3.878431853 4.495536832
[26] 4.891636770 4.321651847 2.586892917 0.287683027 -0.967095231
[31] -0.154801125 1.779184068 3.329748243 3.728816431 3.008381705
[36] 2.062470139 2.040867620 2.344850907 1.972943986 1.507444276
[41] 2.392616997 3.591728656 3.645580543 2.546763564 1.397529938
[46] 1.555075492 2.889476545 4.113456734 4.343448702 3.638212161
[51] 2.760820859 2.177467720 1.847006340 1.845920773 1.768787742
[56] 1.714232294 2.481054630 3.209449345 2.727055649 1.037292802
[61] -0.306982271 0.910333159 3.169372683 4.473565968 4.303055925
[66] 2.680848635 0.091916698 -1.701511677 -1.345663688 -0.677340531
[71] -0.258598841 0.643100195 1.466256774 1.467894896 1.344551184
[76] 1.964176457 2.470124265 2.234811882 1.863764220 2.114591700
[81] 2.736897470 3.270334849 3.616146386 3.732490891 3.475900681
[86] 2.849220489 2.209104569 1.964126712 2.219521464 2.699346344
[91] 2.848076374 2.379471533 1.513906451 0.629701396 0.009238708
[96] 0.104475547 0.934720049 1.812331108 2.350356535 2.594180496
[101] 2.643415190 2.508294240 2.204072185 1.716818077 1.028243611
[106] 0.928492387 2.377011093 3.939163313 4.265217284 3.168610925
[111] 1.305626815 0.192266404 0.045603454 -0.338900123 -0.532965042
[116] 0.706702843 2.088137002 2.216329645 0.896198496 -1.163548677
[121] -2.237262376 -2.070972288 -1.686852188 -0.613200098 0.997570369
[126] 2.061895054 2.658629325 3.542627574 4.271819673 3.804808020
[131] 1.791439737 -0.555095973 -0.300488817 1.136282172 1.824979820
[136] 1.820522589 1.573178989 1.200340261 0.548505678 -0.375932456
[141] -1.592254097 -2.915901616 -2.866964199 -1.257838745 -0.023035779
[146] 0.088847682 -0.775589476 -1.926204672 -2.647717546 -3.191826847
[151] -3.980150578 -4.849247762 -5.414509158 -5.136595392 -4.222254865
[156] -3.807784611 -4.531537440 -6.268987448 -7.749911249 -6.931382998
[161] -4.986319779 -3.802454255 -3.734824180 -4.437747474 -5.440950339
[166] -6.767443015 -8.753589107 -10.690095112 -10.130857997 -8.206825472
[171] -7.008707379 -6.299876627 -5.303776767 -4.050730748 -2.905882278
[176] -2.177646549 -2.202332988 -2.919404859 -3.589023219 -3.877204822
[181] -4.496832829 -5.337874309 -5.005566634 -3.904434399 -3.435418299
[186] -3.458708360 -2.582937024 -0.689832700 0.821690330 1.319743310
[191] 0.917104109 0.153817485 -0.491831027 -1.220225684 -2.360923984
[196] -3.735522536 -5.079883979 -6.297011868 -6.694749698 -6.097206384
[201] -5.430234465 -4.821290729 -4.103499777 -3.622032953 -3.400320177
[206] -2.856772227 -2.019265279 -1.632698722 -2.052250520 -3.067996584
[211] -4.206657227 -5.071861611 -5.327262212 -5.002412420 -4.575575332
[216] -3.932278065 -2.871792959 -2.098864806 -2.192884738 -3.040045898
[221] -4.260774524 -5.646072662 -6.198172847 -4.768791684 -3.083440704
[226] -2.387152874 -2.706874167 -3.490975568 -3.247508272 -1.490320235
[231] 0.078821757 0.647391254 0.364650864 -0.368074036 -1.296252158
[236] -2.441149237 -4.030661342 -6.072433149 -7.233711019 -6.447528473
[241] -5.048214544 -3.619022542 -2.603018471 -2.475305790 -2.888148037
[246] -2.876880004 -2.533773036 -2.672188570 -3.384193293 -4.107360810
[251] -4.263565915 -4.113401271 -4.448918753 -5.626111556 -7.203996858
[256] -7.991906677 -7.517145337 -6.785299581 -6.459353621 -6.370932453
[261] -6.298239645 -6.364846354 -6.408168437 -6.048227736 -5.377868812
[266] -4.660455274 -4.176770101 -4.189093037 -4.619762727 -5.211131228
[271] -5.847187690 -6.145265347 -5.761076297 -5.354122413 -5.563215444
[276] -6.303538852 -7.072500251 -7.467273026 -7.351576861 -6.614122580
[281] -5.372585251 -4.068034813 -2.949534588 -1.978606877 -1.308381743
[286] -1.384066802 -2.523045252 -4.514135926 -6.251465288 -6.818185127
[291] -6.832798720 -6.513652198 -5.431946583 -3.740611314 -2.160250438
[296] -1.343691745 -1.581349923 -2.588984486 -3.443699275 -3.842036150
[301] -4.605381183 -5.909092303 -6.743767149 -6.537929102 -6.077095508
[306] -5.859431994 -5.906986337 -6.174849737 -6.483716974 -6.402675775
[311] -5.805590281 -5.227775087 -5.080566766 -5.256930182 -5.570143348
[316] -6.126470801 -6.799703538 -6.697849566 -5.301495371 -3.467478747
[321] -2.066653629 -1.487221047 -1.901202999 -3.337891643 -5.390581745
[326] -6.904405556 -7.359707930 -7.240370242 -6.443563030 -5.458318602
[331] -5.235776333 -6.173058909 -7.975583726 -9.470520965 -10.016423129
[336] -10.446345364 -10.190023303 -8.384834604 -6.616312467 -5.925647185
[341] -6.263090822 -7.042979212 -7.438112689 -7.129853023 -6.543039365
[346] -6.186880309 -6.214647785 -6.394673536 -6.538363185 -6.541064653
[351] -5.847491815 -4.582458449 -3.845508016 -4.162060413 -5.020303748
[356] -4.964952712 -3.998841526 -3.673752036 -4.639828255 -6.381075286
[361] -7.067875372 -6.202672325 -5.265599723 -4.786713441 -4.642773819
humdiff = artrans.wge (daily_bike_data$ Humidity, phi.tr = c (rep (0 ,364 ),1 ))
aic5.wge (humdiff, type = 'aic' )
---------WORKING... PLEASE WAIT...
Five Smallest Values of aic
p q aic
2 2 5.695175
3 2 5.698350
1 0 5.701318
4 2 5.702775
2 0 5.703852
hum = est.arma.wge (humdiff, p = 2 , q = 2 )
Coefficients of AR polynomial:
-0.5883 0.3890
AR Factor Table
Factor Roots Abs Recip System Freq
1+0.9837B -1.0165 0.9837 0.5000
1-0.3954B 2.5289 0.3954 0.0000
Coefficients of MA polynomial:
-1.0977 -0.1277
MA FACTOR TABLE
Factor Roots Abs Recip System Freq
1+0.9653B -1.0359 0.9653 0.5000
1+0.1323B -7.5572 0.1323 0.5000
acf (hum$ res)
ljung.wge (hum$ res) # Fail to reject
Obs 0.01381014 0.005487669 -0.01488494 -0.09759935 -0.001680299 0.06445373 0.114192 -0.04943437 -0.03187326 -0.01261037 0.03869395 0.04164148 -0.03708437 -0.1178728 0.05037217 0.1033369 -0.02362139 0.00943292 -0.008141213 -0.0451568 -0.0939969 -0.02219895 0.02656584 -0.07296706
$test
[1] "Ljung-Box test"
$K
[1] 24
$chi.square
[1] 30.75562
$df
[1] 24
$pval
[1] 0.1609546
ljung.wge (hum$ res, K= 48 ) # Fail to reject
Obs 0.01381014 0.005487669 -0.01488494 -0.09759935 -0.001680299 0.06445373 0.114192 -0.04943437 -0.03187326 -0.01261037 0.03869395 0.04164148 -0.03708437 -0.1178728 0.05037217 0.1033369 -0.02362139 0.00943292 -0.008141213 -0.0451568 -0.0939969 -0.02219895 0.02656584 -0.07296706 -0.06567389 -0.005756453 0.08350328 0.05608327 -0.01037953 -0.02658103 -0.01989446 -0.02674116 -0.01400777 0.1082012 0.006403354 -0.02056617 -0.07223008 0.08666904 0.0907162 -0.04411947 0.06947638 0.03408037 -0.04054732 0.03166965 0.01998761 0.01298515 -0.006023879 -0.09700959
$test
[1] "Ljung-Box test"
$K
[1] 48
$chi.square
[1] 59.53045
$df
[1] 48
$pval
[1] 0.1228675
preds_Humidity = fore.arima.wge (daily_bike_data$ Humidity, s = 365 , phi = hum$ phi, theta = hum$ theta, n.ahead = 365 )
y.arma -11.33333 -31.47826 0.3977273 -17.58514 8.721014 2.382246 3.297101 -7.083333 26.75 16.36051 16.11364 20.33712 3.708333 -8.032609 -7.958333 3.875 17.8587 -41.83333 -24.42391 -8.833333 37.41667 39.625 47.47283 34.40942 2.679348 -9.291667 5.375 -24.97101 -34.04891 -32.13406 -18.70833 -32.16486 -10.25 8.884058 19.43659 -24.125 5.384058 -24.20833 18.5 6.730072 10.26087 22.48864 -7.958333 -4.61413 13.29167 21.69022 32.94384 12.95833 1.791667 32.79167 10 -1.065217 -0.9861111 13.15399 4.01087 -31.63406 -12.79167 -18.91667 -48.05303 26.97826 16.58333 33.875 1.083333 -38.58333 -44.20109 -9.463768 9.25 -20.79167 40.70833 -29.91486 -11.76268 -3.822464 12.05435 -14.85688 -19.69384 23.91667 23.06159 43.08333 25.5 7.052536 19.66667 -0.8315217 -11.16667 39.04167 48.66667 -1.599638 -1.217391 16.70833 -20.75 -33.75 5.208333 2.25 2.434783 -2.958333 -17.25 -9.666667 -22.58333 -58.20833 -60.16667 -54 -28.19565 -26.96014 -35.29167 -13.20833 -16.83333 -38.04167 8.208333 -15.20833 -9.666667 -0.1666667 28.75 -4.666667 -5.25 -4.416667 -32.25 -30.125 -7.875 -30 3.25 8.375 -19.20833 -7.041667 10 3.125 29.125 16.66667 19.875 3.25 9.708333 25.5 -8.083333 -38.70833 -38.29167 -34.625 -7.75 0.6666667 -14 -35 -30.625 -26.33333 -9.625 6.166667 -4.416667 3.375 2.041667 6.958333 7.875 -3.25 -14.25 -0.04166667 3.333333 -18.41667 45.04167 19.5 3.708333 -16.54167 1.333333 1.333333 -5.5 -10.04167 -16.79167 -11.625 -16 33.875 7.5 9.791667 -9.875 -23.16667 -7.166667 11.125 -5.625 -17.83333 -13.95833 -12.95833 -3.875 -0.4166667 -0.9166667 -28.5 -27.41667 -7.541667 5.458333 20.5 7.416667 -23.54167 -14.58333 -5.166667 -28.54167 -20.04167 -26.58333 -3.541667 10.5 3.166667 7.416667 -10.20833 0.9583333 10.79167 13.29167 4.083333 -14.54167 -7.333333 -10.66667 15.29167 28.5 26.25 14.33333 -10.20833 -9.083333 19.375 1.125 7.083333 15.79167 18.79167 15.33333 18.625 0.2083333 -11.5 -1.75 -10.25 -9.875 11.16667 10.25 19.625 29.20833 31.79167 -19.91667 -27.20833 -2.541667 4.125 -5.625 -8.375 -11.95833 3.75 -3.583333 20.375 22.16667 3.083333 -15.66667 -4.833333 -0.4166667 16.8652 6.541667 0.375 -0.75 -5.166667 -8.875 9.833333 4.875 -3.541667 -14.57065 -10.66667 -20.33152 -9.875 -20.625 -21 -17.21739 -13.54167 -6 -3.666667 -8.875 -14.83333 3.958333 18.25 -34.45833 -28.16667 -23.33333 -32.58333 -39.54167 -35.20833 -27.83333 -25.45833 -15.79167 -0.9166667 -10.45833 -17.04167 -14.25 11.08333 8.375 7.5 0.6666667 -2 0.7083333 -1.791667 2.791667 -17.79167 -44.29167 -35.75 -22.16667 15.70833 22.08333 -2.125 -0.875 -16.68841 17.875 -0.125 -11.91667 -17.29167 -13.04167 1.333333 8 -0.5833333 13.41667 -18.79167 25.625 12.21212 -1.708333 -13.70833 -18 -13.125 1.375 -24.04167 -19.16667 -17.41667 -42.48551 -27.25 19.91667 10.625 28.33333 7.583333 -13.66667 -30.95833 -5.125 13.54167 19.08333 -6.125 -22.5 -34.875 -17.75 1.958333 -23.91667 -21.33333 -16.29167 4.362319 -32.45833 -5.768116 12.5 18.08333 21.04167 -0.8333333 -9.333333 -46.45833 -46.16667 18.41667 21.54167 39.79167 43.5 -7.416667 -5.166667 -17.79167 0.875 15 27.79167 32.08333 2.875 3 -19.04167 -20.08333 -24.5 -2.708333 11 22.78261 6.083333 14.90036 1.583333 11.625 -13.25 -11.5
parzen.wge (daily_bike_data$ Temperature, trunc = 300 )
$freq
[1] 0.001367989 0.002735978 0.004103967 0.005471956 0.006839945 0.008207934
[7] 0.009575923 0.010943912 0.012311902 0.013679891 0.015047880 0.016415869
[13] 0.017783858 0.019151847 0.020519836 0.021887825 0.023255814 0.024623803
[19] 0.025991792 0.027359781 0.028727770 0.030095759 0.031463748 0.032831737
[25] 0.034199726 0.035567715 0.036935705 0.038303694 0.039671683 0.041039672
[31] 0.042407661 0.043775650 0.045143639 0.046511628 0.047879617 0.049247606
[37] 0.050615595 0.051983584 0.053351573 0.054719562 0.056087551 0.057455540
[43] 0.058823529 0.060191518 0.061559508 0.062927497 0.064295486 0.065663475
[49] 0.067031464 0.068399453 0.069767442 0.071135431 0.072503420 0.073871409
[55] 0.075239398 0.076607387 0.077975376 0.079343365 0.080711354 0.082079343
[61] 0.083447332 0.084815321 0.086183311 0.087551300 0.088919289 0.090287278
[67] 0.091655267 0.093023256 0.094391245 0.095759234 0.097127223 0.098495212
[73] 0.099863201 0.101231190 0.102599179 0.103967168 0.105335157 0.106703146
[79] 0.108071135 0.109439124 0.110807114 0.112175103 0.113543092 0.114911081
[85] 0.116279070 0.117647059 0.119015048 0.120383037 0.121751026 0.123119015
[91] 0.124487004 0.125854993 0.127222982 0.128590971 0.129958960 0.131326949
[97] 0.132694938 0.134062927 0.135430917 0.136798906 0.138166895 0.139534884
[103] 0.140902873 0.142270862 0.143638851 0.145006840 0.146374829 0.147742818
[109] 0.149110807 0.150478796 0.151846785 0.153214774 0.154582763 0.155950752
[115] 0.157318741 0.158686731 0.160054720 0.161422709 0.162790698 0.164158687
[121] 0.165526676 0.166894665 0.168262654 0.169630643 0.170998632 0.172366621
[127] 0.173734610 0.175102599 0.176470588 0.177838577 0.179206566 0.180574555
[133] 0.181942544 0.183310534 0.184678523 0.186046512 0.187414501 0.188782490
[139] 0.190150479 0.191518468 0.192886457 0.194254446 0.195622435 0.196990424
[145] 0.198358413 0.199726402 0.201094391 0.202462380 0.203830369 0.205198358
[151] 0.206566347 0.207934337 0.209302326 0.210670315 0.212038304 0.213406293
[157] 0.214774282 0.216142271 0.217510260 0.218878249 0.220246238 0.221614227
[163] 0.222982216 0.224350205 0.225718194 0.227086183 0.228454172 0.229822161
[169] 0.231190150 0.232558140 0.233926129 0.235294118 0.236662107 0.238030096
[175] 0.239398085 0.240766074 0.242134063 0.243502052 0.244870041 0.246238030
[181] 0.247606019 0.248974008 0.250341997 0.251709986 0.253077975 0.254445964
[187] 0.255813953 0.257181943 0.258549932 0.259917921 0.261285910 0.262653899
[193] 0.264021888 0.265389877 0.266757866 0.268125855 0.269493844 0.270861833
[199] 0.272229822 0.273597811 0.274965800 0.276333789 0.277701778 0.279069767
[205] 0.280437756 0.281805746 0.283173735 0.284541724 0.285909713 0.287277702
[211] 0.288645691 0.290013680 0.291381669 0.292749658 0.294117647 0.295485636
[217] 0.296853625 0.298221614 0.299589603 0.300957592 0.302325581 0.303693570
[223] 0.305061560 0.306429549 0.307797538 0.309165527 0.310533516 0.311901505
[229] 0.313269494 0.314637483 0.316005472 0.317373461 0.318741450 0.320109439
[235] 0.321477428 0.322845417 0.324213406 0.325581395 0.326949384 0.328317373
[241] 0.329685363 0.331053352 0.332421341 0.333789330 0.335157319 0.336525308
[247] 0.337893297 0.339261286 0.340629275 0.341997264 0.343365253 0.344733242
[253] 0.346101231 0.347469220 0.348837209 0.350205198 0.351573187 0.352941176
[259] 0.354309166 0.355677155 0.357045144 0.358413133 0.359781122 0.361149111
[265] 0.362517100 0.363885089 0.365253078 0.366621067 0.367989056 0.369357045
[271] 0.370725034 0.372093023 0.373461012 0.374829001 0.376196990 0.377564979
[277] 0.378932969 0.380300958 0.381668947 0.383036936 0.384404925 0.385772914
[283] 0.387140903 0.388508892 0.389876881 0.391244870 0.392612859 0.393980848
[289] 0.395348837 0.396716826 0.398084815 0.399452804 0.400820793 0.402188782
[295] 0.403556772 0.404924761 0.406292750 0.407660739 0.409028728 0.410396717
[301] 0.411764706 0.413132695 0.414500684 0.415868673 0.417236662 0.418604651
[307] 0.419972640 0.421340629 0.422708618 0.424076607 0.425444596 0.426812585
[313] 0.428180575 0.429548564 0.430916553 0.432284542 0.433652531 0.435020520
[319] 0.436388509 0.437756498 0.439124487 0.440492476 0.441860465 0.443228454
[325] 0.444596443 0.445964432 0.447332421 0.448700410 0.450068399 0.451436389
[331] 0.452804378 0.454172367 0.455540356 0.456908345 0.458276334 0.459644323
[337] 0.461012312 0.462380301 0.463748290 0.465116279 0.466484268 0.467852257
[343] 0.469220246 0.470588235 0.471956224 0.473324213 0.474692202 0.476060192
[349] 0.477428181 0.478796170 0.480164159 0.481532148 0.482900137 0.484268126
[355] 0.485636115 0.487004104 0.488372093 0.489740082 0.491108071 0.492476060
[361] 0.493844049 0.495212038 0.496580027 0.497948016 0.499316005
$pzgram
[1] 18.34127014 19.33295660 18.49116970 15.54968535 10.75586223
[6] 6.09680513 4.02057139 3.14092089 2.57247201 1.64560045
[11] 0.77299661 0.65926447 0.75949820 0.19533007 -1.13153453
[16] -1.96766704 -1.41785413 -0.61531045 -0.41015206 -1.03934961
[21] -2.36234340 -3.62515719 -4.16332323 -4.38768691 -3.79951381
[26] -1.97323906 -0.46563877 0.01922867 -0.22841375 -0.91941743
[31] -2.04751273 -3.04466030 -3.11740402 -3.14491297 -3.97557100
[36] -4.93098878 -4.76774497 -4.18747614 -4.48326505 -5.71244648
[41] -6.24247169 -5.32950795 -4.70985239 -4.57353016 -3.96340743
[46] -3.27238705 -3.51753670 -4.69288770 -5.81994149 -6.31909354
[51] -6.21784016 -5.21201990 -4.28908111 -4.40149120 -4.98065530
[56] -4.81695326 -4.54854156 -5.22079546 -6.57252953 -7.35366568
[61] -6.93109722 -6.16060566 -5.95670044 -6.53700722 -7.29818563
[66] -7.40918290 -7.15004734 -7.15224648 -7.31855368 -7.17783796
[71] -6.73686181 -6.50385011 -6.64579756 -6.57334686 -5.99998594
[76] -5.84711666 -6.84637984 -8.16246542 -7.23824341 -5.44466201
[81] -5.02601647 -6.21984146 -7.62243788 -6.82113914 -5.40071306
[86] -4.97200337 -5.31550916 -5.86794443 -6.70449939 -7.94656902
[91] -8.34627845 -7.30946727 -6.33771299 -6.06612761 -6.32254766
[96] -6.51771148 -6.08080782 -5.28932795 -4.54320216 -4.00543709
[101] -3.76310980 -3.41190389 -2.76856819 -2.71594089 -3.99867393
[106] -6.08967529 -6.45194228 -5.20552402 -4.49328166 -4.57384283
[111] -5.12503847 -5.94547698 -6.80800797 -7.27205406 -7.44993272
[116] -7.92151769 -8.42038854 -8.18054811 -7.61920501 -7.34781133
[121] -7.33855819 -7.67761850 -8.47581961 -8.93804340 -8.38058152
[126] -7.89106100 -8.22431462 -8.94781004 -8.74879595 -7.00625409
[131] -5.11875112 -4.30229412 -4.94110126 -7.04349339 -10.18880662
[136] -12.74313944 -12.15909090 -9.07438355 -6.25633537 -4.92084527
[141] -5.29241627 -7.24037521 -10.07454611 -12.28642496 -12.98668920
[146] -12.46472026 -11.58216009 -11.02785250 -10.86405278 -10.99915026
[151] -11.24037987 -11.36250352 -11.50870836 -11.58390007 -10.78204793
[156] -9.44471373 -8.85470303 -9.46410799 -10.47064053 -10.21654245
[161] -9.38254129 -9.39371033 -9.76753540 -9.10398560 -8.19850036
[166] -8.50995220 -10.33107558 -12.46597818 -12.34223547 -11.09211899
[171] -10.70718725 -11.21725097 -11.15848739 -9.90890296 -8.94430275
[176] -8.67994122 -8.64443823 -8.74529067 -9.15031562 -9.60863489
[181] -9.90268951 -10.33500649 -11.22768413 -12.50537351 -13.50282612
[186] -13.57107161 -13.27909418 -13.52063270 -14.49219372 -15.22462315
[191] -14.71037295 -14.17206104 -14.59710262 -15.71638851 -16.76922312
[196] -17.00232811 -15.83270397 -14.20137844 -13.18649623 -12.47869592
[201] -11.40556671 -10.52315088 -10.58322539 -11.26755098 -11.52006185
[206] -11.53384578 -12.31958256 -13.86346458 -15.23501740 -15.97427908
[211] -16.05155167 -15.11941977 -13.75758856 -12.79073460 -12.53549068
[216] -13.07465015 -14.26069628 -15.24841824 -14.97873143 -13.96573122
[221] -12.88416203 -11.80905853 -11.00403851 -10.81378985 -11.39033772
[226] -12.70296637 -14.39261507 -15.61658265 -15.90295025 -15.71405935
[231] -15.54685394 -15.59837943 -15.73590033 -15.59279995 -14.91706817
[236] -14.00057156 -13.42016328 -13.45443658 -14.04269269 -14.98643738
[241] -16.08859676 -16.96899011 -16.85302095 -16.01872104 -15.66260355
[246] -16.02888768 -16.62089143 -17.41712341 -18.83838576 -20.23724704
[251] -20.12590813 -19.40191885 -19.36596724 -19.41924815 -18.17146502
[256] -16.38483976 -15.02173480 -14.05052722 -13.52598339 -13.61134637
[261] -13.99004653 -14.12851539 -14.21670967 -14.61899122 -15.04765678
[266] -15.27106275 -15.54794120 -15.64978445 -15.18504984 -14.46514345
[271] -13.71880522 -13.23589166 -13.65770964 -15.38025268 -18.01436136
[276] -20.00640418 -20.60030105 -20.74584588 -20.79362578 -20.65769737
[281] -20.36184387 -19.75648074 -18.38140046 -16.63420181 -15.41360209
[286] -15.17958020 -16.09525394 -18.14620856 -20.58325199 -20.95975256
[291] -19.26397827 -17.93105324 -17.34992847 -16.58630130 -15.30132418
[296] -14.31988274 -13.92566141 -13.81170309 -13.93688606 -14.58635135
[301] -15.86578973 -17.37999669 -17.69182374 -16.39118710 -15.28887889
[306] -14.90462972 -14.42112223 -13.39650082 -12.53003127 -12.43967309
[311] -13.28613470 -14.90270409 -16.67555961 -17.57969390 -17.14185990
[316] -16.15194517 -15.70700669 -16.32584409 -17.64862297 -18.16190780
[321] -17.35156029 -16.64214661 -16.63859255 -16.91132785 -16.77201943
[326] -16.39993512 -16.48080804 -17.18885223 -18.12497402 -18.86913131
[331] -19.44067062 -19.82217701 -19.81191749 -19.32256117 -18.42644176
[336] -17.64777784 -17.54160898 -18.11343863 -18.92169388 -19.78229851
[341] -21.00798730 -22.06800048 -21.05281603 -18.94525535 -17.67135474
[346] -17.65035831 -18.74869640 -20.35640956 -21.27766197 -21.22908987
[351] -20.83891062 -20.11122540 -19.18831991 -18.41050275 -17.78834832
[356] -17.27847217 -16.82214296 -16.23200552 -15.67545468 -15.67317837
[361] -16.47595256 -17.83681868 -19.31609828 -20.88765969 -22.38148475
tempdiff = artrans.wge (daily_bike_data$ Temperature, phi.tr = c (rep (0 ,364 ),1 ))
plotts.sample.wge (tempdiff)
$xbar
[1] 1.530127
$autplt
[1] 1.000000000 0.672651112 0.303843904 0.137342310 0.135661326
[6] 0.162071006 0.144637010 0.153902985 0.128590825 0.087915982
[11] 0.011149678 -0.044560140 0.003913322 0.063322556 0.098446867
[16] 0.062263260 0.054955303 0.047922129 -0.014798551 -0.030715819
[21] 0.001713817 0.046921231 0.064689447 0.019192525 0.003250627
[26] 0.005805687
$freq
[1] 0.002732240 0.005464481 0.008196721 0.010928962 0.013661202 0.016393443
[7] 0.019125683 0.021857923 0.024590164 0.027322404 0.030054645 0.032786885
[13] 0.035519126 0.038251366 0.040983607 0.043715847 0.046448087 0.049180328
[19] 0.051912568 0.054644809 0.057377049 0.060109290 0.062841530 0.065573770
[25] 0.068306011 0.071038251 0.073770492 0.076502732 0.079234973 0.081967213
[31] 0.084699454 0.087431694 0.090163934 0.092896175 0.095628415 0.098360656
[37] 0.101092896 0.103825137 0.106557377 0.109289617 0.112021858 0.114754098
[43] 0.117486339 0.120218579 0.122950820 0.125683060 0.128415301 0.131147541
[49] 0.133879781 0.136612022 0.139344262 0.142076503 0.144808743 0.147540984
[55] 0.150273224 0.153005464 0.155737705 0.158469945 0.161202186 0.163934426
[61] 0.166666667 0.169398907 0.172131148 0.174863388 0.177595628 0.180327869
[67] 0.183060109 0.185792350 0.188524590 0.191256831 0.193989071 0.196721311
[73] 0.199453552 0.202185792 0.204918033 0.207650273 0.210382514 0.213114754
[79] 0.215846995 0.218579235 0.221311475 0.224043716 0.226775956 0.229508197
[85] 0.232240437 0.234972678 0.237704918 0.240437158 0.243169399 0.245901639
[91] 0.248633880 0.251366120 0.254098361 0.256830601 0.259562842 0.262295082
[97] 0.265027322 0.267759563 0.270491803 0.273224044 0.275956284 0.278688525
[103] 0.281420765 0.284153005 0.286885246 0.289617486 0.292349727 0.295081967
[109] 0.297814208 0.300546448 0.303278689 0.306010929 0.308743169 0.311475410
[115] 0.314207650 0.316939891 0.319672131 0.322404372 0.325136612 0.327868852
[121] 0.330601093 0.333333333 0.336065574 0.338797814 0.341530055 0.344262295
[127] 0.346994536 0.349726776 0.352459016 0.355191257 0.357923497 0.360655738
[133] 0.363387978 0.366120219 0.368852459 0.371584699 0.374316940 0.377049180
[139] 0.379781421 0.382513661 0.385245902 0.387978142 0.390710383 0.393442623
[145] 0.396174863 0.398907104 0.401639344 0.404371585 0.407103825 0.409836066
[151] 0.412568306 0.415300546 0.418032787 0.420765027 0.423497268 0.426229508
[157] 0.428961749 0.431693989 0.434426230 0.437158470 0.439890710 0.442622951
[163] 0.445355191 0.448087432 0.450819672 0.453551913 0.456284153 0.459016393
[169] 0.461748634 0.464480874 0.467213115 0.469945355 0.472677596 0.475409836
[175] 0.478142077 0.480874317 0.483606557 0.486338798 0.489071038 0.491803279
[181] 0.494535519 0.497267760 0.500000000
$dbz
[1] 6.86101403 6.80876323 6.72324697 6.60681842 6.46275280
[6] 6.29518360 6.10896488 5.90943557 5.70207779 5.49209097
[11] 5.28393959 5.08096222 4.88513693 4.69707086 4.51622501
[16] 4.34131772 4.17080026 4.00328139 3.83779884 3.67388279
[21] 3.51141617 3.35035329 3.19040163 3.03078743 2.87020901
[26] 2.70703353 2.53972834 2.36745799 2.19073589 2.01199999
[31] 1.83597637 1.66969949 1.52208037 1.40297641 1.32183779
[36] 1.28616944 1.30018185 1.36400724 1.47368247 1.62182413
[41] 1.79870088 1.99335525 2.19452831 2.39129863 2.57346871
[46] 2.73178552 2.85808032 2.94538703 2.98807204 2.98199008
[51] 2.92467221 2.81554605 2.65618008 2.45052682 2.20511017
[56] 1.92906089 1.63386261 1.33265826 1.03902624 0.76530524
[61] 0.52079716 0.31038312 0.13406916 -0.01233017 -0.13560508
[66] -0.24320235 -0.34147694 -0.43450667 -0.52359030 -0.60738021
[71] -0.68252028 -0.74462569 -0.78941644 -0.81379494 -0.81666574
[76] -0.79935507 -0.76559578 -0.72115865 -0.67328426 -0.63007626
[81] -0.59996988 -0.59132308 -0.61212267 -0.66976563 -0.77086635
[86] -0.92104105 -1.12462508 -1.38428033 -1.70045122 -2.07063612
[91] -2.48847413 -2.94273603 -3.41648838 -3.88697956 -4.32706285
[96] -4.70888160 -5.00963751 -5.21760888 -5.33545222 -5.37882273
[101] -5.37112173 -5.33721388 -5.29867154 -5.27146464 -5.26568972
[106] -5.28650316 -5.33555591 -5.41248757 -5.51622460 -5.64592813
[111] -5.80149774 -5.98359790 -6.19324142 -6.43102448 -6.69614549
[116] -6.98535979 -7.29204724 -7.60562670 -7.91163947 -8.19287331
[121] -8.43176420 -8.61384892 -8.73133573 -8.78539396 -8.78610788
[126] -8.75015225 -8.69730281 -8.64713015 -8.61669081 -8.61932215
[131] -8.66421416 -8.75632540 -8.89629021 -9.08009989 -9.29849275
[136] -9.53618459 -9.77136785 -9.97629436 -10.12000705 -10.17384969
[141] -10.11878205 -9.95144149 -9.68550349 -9.34748397 -8.96955464
[146] -8.58297237 -8.21412579 -7.88318277 -7.60436548 -7.38689424
[151] -7.23600789 -7.15379983 -7.13980723 -7.19138491 -7.30393449
[156] -7.47107535 -7.68485057 -7.93605820 -8.21477424 -8.51107680
[161] -8.81588863 -9.12175202 -9.42328792 -9.71712517 -10.00122947
[166] -10.27376628 -10.53181430 -10.77034554 -10.98188641 -11.15716066
[171] -11.28676196 -11.36350549 -11.38469739 -11.35342402 -11.27831471
[176] -11.17192568 -11.04846399 -10.92167297 -10.80339363 -10.70290660
[181] -10.62688205 -10.57966869 -10.56367547
aic5.wge (tempdiff, type = 'bic' ) # bic picked p = 1 and q = 1
---------WORKING... PLEASE WAIT...
Five Smallest Values of bic
p q bic
1 1 3.714381
2 0 3.720397
3 0 3.722154
0 2 3.723965
1 2 3.726426
temp = est.arma.wge (tempdiff, p = 1 , q = 1 )
Coefficients of AR polynomial:
0.4839
AR Factor Table
Factor Roots Abs Recip System Freq
1-0.4839B 2.0667 0.4839 0.0000
Coefficients of MA polynomial:
-0.3906
MA FACTOR TABLE
Factor Roots Abs Recip System Freq
1+0.3906B -2.5605 0.3906 0.5000
ljung.wge (temp$ res) # close to 0.05 but Fail to reject
Obs 0.01228195 -0.0002127207 -0.07945629 0.0267131 0.1107299 -0.02603524 0.1025488 0.01861338 0.05959623 0.01229143 -0.1240381 0.03700199 0.004279513 0.115961 -0.03897411 0.0003535096 0.08749307 -0.05580738 -0.03367594 -0.0009418035 0.01374499 0.088237 -0.05257655 0.005048402
$test
[1] "Ljung-Box test"
$K
[1] 24
$chi.square
[1] 33.87608
$df
[1] 24
$pval
[1] 0.08689569
ljung.wge (temp$ res, K= 48 ) # Fail to reject
Obs 0.01228195 -0.0002127207 -0.07945629 0.0267131 0.1107299 -0.02603524 0.1025488 0.01861338 0.05959623 0.01229143 -0.1240381 0.03700199 0.004279513 0.115961 -0.03897411 0.0003535096 0.08749307 -0.05580738 -0.03367594 -0.0009418035 0.01374499 0.088237 -0.05257655 0.005048402 -0.008982536 0.01088205 -0.008136983 0.004781849 0.08392675 -0.0567659 -0.02341514 -0.05085354 0.03217923 -0.04656861 -0.0141383 0.02833786 0.05223805 -0.002588415 -0.02656257 -0.04648866 0.04865447 0.002639143 0.04907394 0.04937573 0.1014075 0.006801172 0.01647366 -0.07264651
$test
[1] "Ljung-Box test"
$K
[1] 48
$chi.square
[1] 53.06362
$df
[1] 48
$pval
[1] 0.2852875
preds_Temperature = fore.arima.wge (daily_bike_data$ Temperature, s = 365 , phi = temp$ phi, theta = temp$ theta, n.ahead = 365 )
y.arma 2.229167 -7.821739 -3.998106 -7.997645 3.351449 11.22011 17.02029 14.90417 7.408333 13.62971 9.052273 18.11364 9.408333 1.66413 -5.758333 -3.595833 17.04112 7.466667 -8.842029 -3.8125 -0.3666667 8.934239 10.49529 21.15018 6.088406 10.68542 19.84583 9.702717 7.414674 4.530072 18.05 23.91775 11.99583 10.91649 4.561594 2.804167 -0.2550725 7.133333 3.091667 11.23261 11.77355 3.0375 -8.191667 -8.112138 -8.279167 7.09837 -0.1268116 -7.991667 -15.09583 -10.29167 -0.4521739 -1.332609 18.44444 20.08279 9.667029 -6.338587 -0.2916667 2.007246 -4.160985 6.703986 13.0125 13.4 13.175 -5.0375 -11.49891 -0.2813406 9.629167 20.00833 1.868939 -2.495833 2.812681 7.093116 20.70199 22.01304 16.58279 1.804167 -2.232971 0 18.35 11.26957 7.770833 17.87808 27.325 20.5875 14.825 16.64928 5.100181 15.69167 16.7625 8.783333 10.725 9.575 4.801449 -9.2125 11 3.804167 -2.954167 8.770833 13.575 5.3875 -12.85815 -13.27609 -1.304167 -2.1625 4.166667 15.15833 17.90833 8.279167 -3.683333 -8.354167 5.825 20.1375 -5.441667 -22.425 -16.69167 -13.375 -10.48333 -13.79583 -11.50833 -1.225 1.091667 5.5375 -4.5375 12.6 14.5375 12.27917 3.6875 0.7916667 4.241667 3.654167 -3.1625 -0.1458333 4.441667 7.916667 0.925 2.941667 6.45 3.725 2.8875 5.458333 1.570833 -0.5083333 -1.445833 -3.308333 -0.5 -2.441667 0.9375 2.958333 3.8875 -0.9208333 -10.22083 -7.283333 -5.245833 -3.170833 -2.808333 -4.395833 -11.85833 -13.22917 -14.9625 -13.74167 -3.808333 0.1458333 2.4375 1.583333 4.458333 1.870833 0.9333333 -1.525 -9.004167 -11.28333 4.616667 8.7625 6.2625 4.258333 0.6583333 4.170833 3.075 -4.466667 -4.020833 1.791667 11.875 3.654167 6.7 5.625 4.6875 3.679167 9.279167 6.775 13.175 7.7 -3.170833 -3.595833 -6.7125 -2.679167 4.383333 3.441667 5.095833 3.820833 6.183333 1.45 0.1333333 -12.90417 -21.79167 -15.68333 -7.633333 0.6458333 -4.1125 0.125 0.2208333 -7.125 -7.145833 -6.495833 -5.045833 -5.6875 1.775 4.816667 7.125 4.541667 0.8708333 -2.520833 -2.15 -0.9583333 -0.1333333 -1.3875 1.291667 3.808333 5.25 0.5041667 -0.35 1.025 -0.5666667 -5.3375 -6.4625 -3.670833 2.3125 1.941667 1.583333 -3.329167 -2.298611 -0.322549 7.9125 3.954167 4.308333 9.433333 9.5 2.379167 -0.1291667 4.541667 16.96667 8.408333 5.987862 0.7708333 -4.320833 -6.045833 -5.770471 -4.466667 -5.25 4.8125 12.01667 7.595833 6.320833 6.416667 -0.7791667 -4.170833 -2.520833 3.508333 -6.6625 -10.35417 -8.558333 -0.1375 1.283333 0.2041667 -1.879167 10.0625 14.16667 17.82917 14.9625 10.26667 10.43333 3.754167 -9.125 -13.6125 -10.7125 -4.541667 -9.35 -13.07917 -13.5875 1.283333 4.320833 -5.666667 -6.620833 -1.656159 7.55 4.8875 3.591667 5.683333 6.983333 9.983333 5.691667 6.5625 17.17083 19.25417 10.46667 -1.869697 -3.7375 -1.004167 -4.6 -5.170833 -0.06666667 -2.508333 -9.845833 -9.733333 -4.12971 -1.608333 5.616667 5.545833 3.808333 -16.11667 -20.775 -11.65833 0.3041667 4.391667 1.1625 -7.1 -6.3375 -5.483333 -8.7 -0.4375 -8.345833 -11.21667 -12.59583 -18.28496 -13.9625 -3.807246 -1.216667 -1.370833 4.1625 10.50833 7.75 -2.091667 -13.31667 4.766667 7.85 9.454167 18.56667 9.9375 1.2875 -1.875 -12.17083 -4.379167 9.029167 13.39167 11.58333 -4.6375 -8.495833 -8.3625 -9.295833 -4.891667 -3.756522 -2.617391 -7.0625 -3.8875 0.45 -5.045833 -13.325 -13.32083
sequence_length <- 366
pred_Days_of_the_Week <- (2 : sequence_length) %% 7
plotts.sample.wge (daily_bike_data$ Wind_Speed, trunc = 300 )
$xbar
[1] 12.76227
$autplt
[1] 1.000000000 0.326423464 0.056437335 0.055355010 0.041954482
[6] 0.078641959 0.063566217 0.012863218 -0.011895845 0.033532168
[11] 0.094405045 0.102575797 0.034418821 0.070151667 0.072308807
[16] 0.115595551 0.172187129 0.063167847 0.032315535 -0.002191299
[21] -0.030467438 0.030991528 0.019502839 -0.013835894 0.034480516
[26] 0.041820645
$freq
[1] 0.001367989 0.002735978 0.004103967 0.005471956 0.006839945 0.008207934
[7] 0.009575923 0.010943912 0.012311902 0.013679891 0.015047880 0.016415869
[13] 0.017783858 0.019151847 0.020519836 0.021887825 0.023255814 0.024623803
[19] 0.025991792 0.027359781 0.028727770 0.030095759 0.031463748 0.032831737
[25] 0.034199726 0.035567715 0.036935705 0.038303694 0.039671683 0.041039672
[31] 0.042407661 0.043775650 0.045143639 0.046511628 0.047879617 0.049247606
[37] 0.050615595 0.051983584 0.053351573 0.054719562 0.056087551 0.057455540
[43] 0.058823529 0.060191518 0.061559508 0.062927497 0.064295486 0.065663475
[49] 0.067031464 0.068399453 0.069767442 0.071135431 0.072503420 0.073871409
[55] 0.075239398 0.076607387 0.077975376 0.079343365 0.080711354 0.082079343
[61] 0.083447332 0.084815321 0.086183311 0.087551300 0.088919289 0.090287278
[67] 0.091655267 0.093023256 0.094391245 0.095759234 0.097127223 0.098495212
[73] 0.099863201 0.101231190 0.102599179 0.103967168 0.105335157 0.106703146
[79] 0.108071135 0.109439124 0.110807114 0.112175103 0.113543092 0.114911081
[85] 0.116279070 0.117647059 0.119015048 0.120383037 0.121751026 0.123119015
[91] 0.124487004 0.125854993 0.127222982 0.128590971 0.129958960 0.131326949
[97] 0.132694938 0.134062927 0.135430917 0.136798906 0.138166895 0.139534884
[103] 0.140902873 0.142270862 0.143638851 0.145006840 0.146374829 0.147742818
[109] 0.149110807 0.150478796 0.151846785 0.153214774 0.154582763 0.155950752
[115] 0.157318741 0.158686731 0.160054720 0.161422709 0.162790698 0.164158687
[121] 0.165526676 0.166894665 0.168262654 0.169630643 0.170998632 0.172366621
[127] 0.173734610 0.175102599 0.176470588 0.177838577 0.179206566 0.180574555
[133] 0.181942544 0.183310534 0.184678523 0.186046512 0.187414501 0.188782490
[139] 0.190150479 0.191518468 0.192886457 0.194254446 0.195622435 0.196990424
[145] 0.198358413 0.199726402 0.201094391 0.202462380 0.203830369 0.205198358
[151] 0.206566347 0.207934337 0.209302326 0.210670315 0.212038304 0.213406293
[157] 0.214774282 0.216142271 0.217510260 0.218878249 0.220246238 0.221614227
[163] 0.222982216 0.224350205 0.225718194 0.227086183 0.228454172 0.229822161
[169] 0.231190150 0.232558140 0.233926129 0.235294118 0.236662107 0.238030096
[175] 0.239398085 0.240766074 0.242134063 0.243502052 0.244870041 0.246238030
[181] 0.247606019 0.248974008 0.250341997 0.251709986 0.253077975 0.254445964
[187] 0.255813953 0.257181943 0.258549932 0.259917921 0.261285910 0.262653899
[193] 0.264021888 0.265389877 0.266757866 0.268125855 0.269493844 0.270861833
[199] 0.272229822 0.273597811 0.274965800 0.276333789 0.277701778 0.279069767
[205] 0.280437756 0.281805746 0.283173735 0.284541724 0.285909713 0.287277702
[211] 0.288645691 0.290013680 0.291381669 0.292749658 0.294117647 0.295485636
[217] 0.296853625 0.298221614 0.299589603 0.300957592 0.302325581 0.303693570
[223] 0.305061560 0.306429549 0.307797538 0.309165527 0.310533516 0.311901505
[229] 0.313269494 0.314637483 0.316005472 0.317373461 0.318741450 0.320109439
[235] 0.321477428 0.322845417 0.324213406 0.325581395 0.326949384 0.328317373
[241] 0.329685363 0.331053352 0.332421341 0.333789330 0.335157319 0.336525308
[247] 0.337893297 0.339261286 0.340629275 0.341997264 0.343365253 0.344733242
[253] 0.346101231 0.347469220 0.348837209 0.350205198 0.351573187 0.352941176
[259] 0.354309166 0.355677155 0.357045144 0.358413133 0.359781122 0.361149111
[265] 0.362517100 0.363885089 0.365253078 0.366621067 0.367989056 0.369357045
[271] 0.370725034 0.372093023 0.373461012 0.374829001 0.376196990 0.377564979
[277] 0.378932969 0.380300958 0.381668947 0.383036936 0.384404925 0.385772914
[283] 0.387140903 0.388508892 0.389876881 0.391244870 0.392612859 0.393980848
[289] 0.395348837 0.396716826 0.398084815 0.399452804 0.400820793 0.402188782
[295] 0.403556772 0.404924761 0.406292750 0.407660739 0.409028728 0.410396717
[301] 0.411764706 0.413132695 0.414500684 0.415868673 0.417236662 0.418604651
[307] 0.419972640 0.421340629 0.422708618 0.424076607 0.425444596 0.426812585
[313] 0.428180575 0.429548564 0.430916553 0.432284542 0.433652531 0.435020520
[319] 0.436388509 0.437756498 0.439124487 0.440492476 0.441860465 0.443228454
[325] 0.444596443 0.445964432 0.447332421 0.448700410 0.450068399 0.451436389
[331] 0.452804378 0.454172367 0.455540356 0.456908345 0.458276334 0.459644323
[337] 0.461012312 0.462380301 0.463748290 0.465116279 0.466484268 0.467852257
[343] 0.469220246 0.470588235 0.471956224 0.473324213 0.474692202 0.476060192
[349] 0.477428181 0.478796170 0.480164159 0.481532148 0.482900137 0.484268126
[355] 0.485636115 0.487004104 0.488372093 0.489740082 0.491108071 0.492476060
[361] 0.493844049 0.495212038 0.496580027 0.497948016 0.499316005
$dbz
[1] 8.177218907 8.749359188 7.905939871 5.681016856 3.064792060
[6] 1.164661410 -0.142763771 -0.763396339 -0.481273119 0.317064061
[11] 1.213786007 2.102627933 3.156880867 4.112308710 4.417826208
[16] 3.873956802 2.721370972 1.612814367 1.362936809 1.802493360
[21] 1.882963138 1.007338804 -0.677142701 -2.195041318 -2.446326027
[26] -1.777441319 -0.941487814 -0.178964193 0.350705804 0.580007671
[31] 0.880302070 1.418940310 1.496226837 0.585276909 -0.489599601
[36] -0.033622803 0.990426800 1.297202499 1.103770499 0.886890880
[41] 0.627062126 0.455099549 0.999754169 2.221921558 3.205587451
[46] 3.506806305 3.614163672 4.082407319 4.439030630 4.204519941
[51] 3.726989501 3.699723825 4.106866900 4.252430511 3.698240741
[56] 2.820840910 2.638217662 3.125391761 3.309702653 2.683930292
[61] 1.157133068 -0.898236060 -2.043566942 -1.630908107 -1.362822991
[66] -1.918238270 -2.137305102 -0.945885729 0.723002036 2.075879307
[71] 2.664907738 2.038850287 0.005499153 -2.506382522 -2.561634541
[76] -0.674260202 0.832347695 1.328259716 0.816982582 -0.151508951
[81] -0.929239589 -1.604494148 -1.576415831 0.143075265 2.078264462
[86] 3.146851891 3.409384556 3.211348955 2.783101306 2.104342192
[91] 1.010089587 -0.263800693 -0.400220702 0.751719483 1.455837263
[96] 1.055963854 0.269870399 0.606887841 1.823825122 2.566536079
[101] 2.409234159 1.608237637 0.601391186 -0.482343799 -1.365658626
[106] -1.086777662 0.019694441 0.626827392 0.223545533 -0.955916969
[111] -1.954280645 -2.095280525 -2.260376630 -3.114778465 -3.723867263
[116] -2.712108931 -1.002775130 0.411958172 1.277335698 1.366348542
[121] 0.481964436 -1.309409906 -2.592442146 -0.892813992 1.456771248
[126] 2.491974152 2.304901559 1.736222868 1.584378852 1.971756505
[131] 2.625866533 2.960023501 2.707236269 2.342333213 2.248129265
[136] 2.262255562 2.514493561 2.927416854 2.972047389 2.619149852
[141] 2.289999038 1.964365674 1.350132986 1.139888218 2.287100991
[146] 3.534425882 3.596606465 2.307599663 0.405690384 -0.697793783
[151] -0.713726560 -0.222555737 0.440245631 0.862018844 0.818965256
[156] 0.449221824 0.096793225 -0.100968167 -0.386951946 -0.942344972
[161] -1.635047628 -2.136114375 -2.110003448 -1.594835659 -1.008585472
[166] -0.747035524 -0.950533109 -1.229117240 -1.016495098 -0.613646926
[171] -0.506214015 -0.572474643 -0.712955511 -1.165180541 -1.777377289
[176] -1.805612328 -0.996320951 0.047395837 0.626836518 0.401674968
[181] -0.295501685 -0.661760737 -0.713861076 -1.087246343 -1.643429804
[186] -1.548915677 -0.883928592 -0.723533924 -1.639221446 -3.225654112
[191] -4.107412815 -3.817234945 -3.031117591 -2.073936260 -1.597412452
[196] -2.055047754 -2.541965011 -1.640140174 -0.515222140 -0.296789856
[201] -0.703997618 -1.076168949 -1.172940393 -0.717652709 0.416428618
[206] 1.391977859 1.357897033 0.025119508 -1.823955428 -1.859005725
[211] -0.830088046 -0.835265190 -1.838239414 -2.564308991 -2.398803526
[216] -2.027441349 -1.787697865 -1.774450427 -1.864520841 -1.513503369
[221] -0.767121654 -0.426354849 -0.730276205 -0.937734836 -0.594313193
[226] -0.521629272 -1.190729274 -1.920104380 -1.574175311 -0.444747337
[231] 0.380062674 0.630592384 0.688906478 0.533950570 -0.345850215
[236] -1.567439017 -1.714895255 -1.387416208 -1.887065108 -2.796779365
[241] -2.443932822 -0.890161724 0.109746704 -0.297665294 -1.899066890
[246] -2.812607666 -1.980080873 -1.406365941 -1.836025486 -2.691765909
[251] -3.157233507 -3.548357996 -4.901515673 -7.719295128 -11.007119421
[256] -10.298468594 -6.590835470 -3.920808460 -2.972214471 -3.428137600
[261] -4.312533553 -4.391226454 -3.623790158 -2.741957742 -2.026558645
[266] -1.265947134 -0.465198494 0.050683435 0.220345545 0.212540605
[271] 0.294382421 0.756740396 1.202995973 0.843952570 -0.538434444
[276] -2.109072880 -2.491856333 -2.222949687 -2.535181418 -3.248694884
[281] -2.917086421 -1.533555649 -0.422332668 -0.139979873 -0.818740158
[286] -2.181488413 -3.236556641 -3.326732282 -2.736373448 -1.950808606
[291] -1.881501312 -3.185991085 -5.580478771 -6.870828723 -5.908706086
[296] -4.968152342 -5.061797043 -6.080968876 -7.187844425 -7.199543533
[301] -6.030620626 -4.709887134 -4.065380853 -4.266674262 -4.919268279
[306] -5.329837325 -5.106873057 -4.426711585 -3.370047088 -1.901752164
[311] -0.693279709 -0.594805767 -1.926432377 -4.025965856 -4.887550845
[316] -4.605855728 -4.739337199 -4.636484646 -3.060738723 -1.290692719
[321] -0.676748554 -1.535933325 -3.581108675 -5.288342933 -4.900284716
[326] -3.920178311 -3.582816936 -3.600941089 -3.390011523 -3.036689889
[331] -2.734068255 -2.299763831 -1.904669235 -2.033559177 -2.767826598
[336] -3.670065546 -4.358391479 -5.004743902 -5.619481217 -5.384501876
[341] -4.066957111 -2.817161605 -2.409390771 -2.855869701 -3.655714115
[346] -4.234687536 -4.590581199 -5.062124029 -5.877411226 -6.948512770
[351] -7.698746719 -7.987072780 -7.969760413 -6.656336188 -4.607166429
[356] -3.360395072 -3.160854551 -3.599473823 -4.396389950 -5.445097395
[361] -5.812084798 -4.813891817 -3.834628183 -3.740777778 -4.154685737
winddiff = artrans.wge (daily_bike_data$ Wind_Speed, phi.tr = c (rep (0 ,364 ),1 ))
aic5.wge (winddiff, type = 'bic' ) # bic and aic picked p = 0 and q = 1
---------WORKING... PLEASE WAIT...
Five Smallest Values of bic
p q bic
0 1 3.908788
1 0 3.910454
1 1 3.924214
0 2 3.924318
2 0 3.924622
wind = est.arma.wge (winddiff, p = 0 , q = 1 )
acf (wind$ res)
ljung.wge (wind$ res) # Fail to reject
Obs 0.007531209 0.02983954 0.01709315 -0.03201086 -0.06295044 -0.03242629 0.01186228 -0.09980544 -0.0365967 -0.004443714 0.02623119 -0.0620031 -0.01971814 0.03985224 0.0187813 0.1125395 -0.006269405 -0.02688986 -0.04856493 -0.09592838 0.01991974 -0.02800076 -0.1347774 0.06464082
$test
[1] "Ljung-Box test"
$K
[1] 24
$chi.square
[1] 28.54717
$df
[1] 24
$pval
[1] 0.2376395
ljung.wge (wind$ res, K= 48 ) # Fail to reject
Obs 0.007531209 0.02983954 0.01709315 -0.03201086 -0.06295044 -0.03242629 0.01186228 -0.09980544 -0.0365967 -0.004443714 0.02623119 -0.0620031 -0.01971814 0.03985224 0.0187813 0.1125395 -0.006269405 -0.02688986 -0.04856493 -0.09592838 0.01991974 -0.02800076 -0.1347774 0.06464082 -0.03134292 0.02687275 0.01003496 0.03298694 0.03970051 -0.001701995 -0.04168581 0.002833127 0.02720862 -0.09361458 0.04431705 0.01422789 -0.0215587 0.04944114 -0.04842808 0.0466781 0.01307725 0.08944093 0.01784868 0.05844817 0.02940169 -0.01060349 -0.06301124 -0.02997102
$test
[1] "Ljung-Box test"
$K
[1] 48
$chi.square
[1] 46.24967
$df
[1] 48
$pval
[1] 0.5448044
preds_Wind_Speed = fore.arima.wge (daily_bike_data$ Wind_Speed, s = 365 , phi = wind$ phi, theta = wind$ theta, n.ahead = 365 )
y.arma 2.125 5.434783 7.863636 1.63587 -3.813406 5.25 0.4039855 -5.041667 -17.625 -2.393116 0.6098485 -8.284091 5.166667 4.063406 6.25 2.875 10.3913 18 0.7934783 0.4583333 -8.75 1.853261 -9.105072 -2.317029 2.096014 -14.77083 15.33333 5.86413 6.344203 9.501812 5 9.101449 -5.166667 -6.650362 -0.3985507 1 0.8043478 6.833333 -15.29167 0.3478261 -7.036232 12.14394 13.79167 -6.269928 -18.54167 -7.355072 -10.74457 -1.625 -4.958333 -17.04167 0.3913043 -6.842391 2.597222 6.487319 -0.865942 5.032609 1.25 9.567029 -6.450758 -2.5 -5.416667 -5.416667 -2.833333 5.541667 -7.666667 -9.411232 15.08333 14.83333 10.24621 -0.4836957 0.1213768 -4.213768 6.701087 -4.639493 -3.567029 -6.458333 -8.061594 -16.16667 -3.041667 -11.22283 -9.125 -7.820652 -8.541667 -2.541667 0.75 13.56884 -1.675725 4.333333 9.833333 -5.333333 -0.5416667 -1.666667 8.70471 -19.125 -13.875 -2.916667 9.25 3.208333 6.666667 14.16667 -5.030797 1.365942 2.666667 3 -2.375 -7.75 -1.25 7.375 0.6666667 -11.79167 -11.75 4.291667 7.625 7.458333 4.291667 -13.91667 -9.083333 1.791667 -7.375 -7.916667 4.375 -1.833333 -13.70833 -13.08333 -8.875 -5.041667 -0.7083333 10.45833 8.041667 6.75 12.95833 3.166667 -3.791667 6.041667 4 1.375 -10.375 1.875 1.916667 -2.75 8.958333 5.708333 -7.75 -7.041667 1.25 -3.958333 -2.791667 -1 -1.166667 11 1.541667 -0.7916667 -3.666667 -4.458333 4.083333 9.791667 5.875 -7.416667 1.375 1.708333 0.25 -1.416667 2.958333 -6.083333 4.958333 5.75 -2 1.583333 1.708333 4.875 -0.5 -3.833333 -3.583333 -3.75 -2.833333 -4.291667 13.79167 16.08333 8.541667 -6.041667 -1.333333 3.916667 3.583333 -2.25 3 0.5833333 2.958333 -0.875 -4.125 -2.833333 -0.1666667 -8.75 -3.25 0.04166667 -10.70833 -2.625 -2.791667 -5.416667 -6.791667 0.4583333 3.458333 -0.875 5.375 -2.5 -2.041667 8.083333 -2.375 6.791667 -1.708333 -1.125 0.125 -0.75 0.5833333 -4.333333 -0.4166667 1.25 4.875 4.166667 -4.833333 -5.041667 -2.708333 -2.958333 4.958333 5.416667 -5.958333 -5.791667 -2.666667 -4.458333 -0.125 -0.125 2.583333 -1.208333 -7.958333 -13.66667 -5.125 -11.29167 -9.291667 10.75 -9.833333 -11.78676 2.083333 -0.8333333 -0.4166667 1.791667 -1.791667 -8.125 -3.708333 1.583333 -10.46014 3.041667 -1.413043 10.5 4.75 9.625 0.2101449 -0.6666667 -5.666667 -11.20833 5.583333 -6.666667 -1.791667 13.79167 5.375 1.458333 1.75 13.75 9.708333 6.166667 8.416667 8.458333 -0.9583333 -0.5833333 1.416667 -10.54167 -8.791667 1.416667 -9.333333 -4 -1.916667 16.45833 6.416667 8.458333 9.916667 3 -4.458333 6.25 -5.208333 1.375 0.9583333 0.4583333 -0.5833333 -0.4293478 -19.25 -6.958333 4.916667 -1.208333 -1.625 -6.291667 -1.625 -4.375 0.4166667 3.125 12.16667 7.147727 2.083333 5.041667 8.666667 -0.08333333 -0.6666667 9.666667 7.791667 15.75 19.13768 1.708333 -17.20833 -5.666667 -7.25 2.375 0 1.083333 -9 0.75 0.2083333 3.291667 -3.708333 -1.041667 -18.95833 -1.291667 18.625 5.5 -10.875 6.394928 -3.166667 -10.42754 -10.83333 -2.75 1.916667 -0.08333333 7.5 6.125 -6.125 -7.333333 1.25 -5.041667 8.291667 15.58333 1.5 7.583333 -9.166667 -10.33333 -9.541667 -4.791667 3.25 8.208333 -5.958333 21.91667 8.916667 -3.833333 -5.217391 -4.73913 8.583333 3.762681 2.416667 -0.6666667 8.75 -2.5
ahead = cbind (Humidity = preds_Humidity$ f,Temperature = preds_Temperature$ f,Days_of_the_Week = pred_Days_of_the_Week, Wind_Speed = preds_Wind_Speed$ f)
fit3 = lm (Total_Users~ Humidity + Temperature + Day_of_the_Week + Wind_Speed, data = daily_bike_data)
aic5.wge (fit3$ residuals, type = 'bic' )
---------WORKING... PLEASE WAIT...
Five Smallest Values of bic
p q bic
2 1 13.33838
3 1 13.34294
4 1 13.34794
3 2 13.35016
1 2 13.35355
aic5.wge (fit3$ residuals, type = 'aic' )
---------WORKING... PLEASE WAIT...
Five Smallest Values of aic
p q aic
4 1 13.31023
3 1 13.31151
5 1 13.31174
3 2 13.31245
2 1 13.31324
mlr_forecast = arima (daily_bike_data$ Total_Users, order = c (2 ,0 ,1 ),xreg = cbind (daily_bike_data$ Humidity, daily_bike_data$ Temperature, daily_bike_data$ Day_of_the_Week, daily_bike_data$ Wind_Speed))
plotts.wge (mlr_forecast$ residuals) # looks random
acf (mlr_forecast$ residuals) # 0/20 acfs out of bounds
ljung.wge (mlr_forecast$ residuals) # greater than 0.05
Obs -0.01819311 0.06316353 -0.04319756 -0.03826881 -0.02349991 0.0493189 0.005386681 0.006631483 0.03295155 0.03328202 -0.02737098 -0.06456494 -0.03694843 0.003360541 0.04931192 -0.05314118 0.03200981 -0.06984514 -0.002852728 0.01643167 -0.01610992 -0.03119754 -0.02401377 -0.04086988
$test
[1] "Ljung-Box test"
$K
[1] 24
$chi.square
[1] 25.42236
$df
[1] 24
$pval
[1] 0.3831083
ljung.wge (mlr_forecast$ residuals, K = 48 )
Obs -0.01819311 0.06316353 -0.04319756 -0.03826881 -0.02349991 0.0493189 0.005386681 0.006631483 0.03295155 0.03328202 -0.02737098 -0.06456494 -0.03694843 0.003360541 0.04931192 -0.05314118 0.03200981 -0.06984514 -0.002852728 0.01643167 -0.01610992 -0.03119754 -0.02401377 -0.04086988 -0.03952179 0.03528498 0.04641581 0.045521 0.09108643 -0.04627283 -0.004823374 -0.02828935 -0.02774926 0.0104396 0.01759409 -0.007428014 0.09276149 -0.06989438 -0.0007048291 -0.004138593 -0.01376151 0.008681763 0.00441694 -0.03534546 -0.02328947 0.02102787 0.003389228 0.00107699
$test
[1] "Ljung-Box test"
$K
[1] 48
$chi.square
[1] 52.73875
$df
[1] 48
$pval
[1] 0.2959252
mlr_st_pred2 = predict (mlr_forecast, newxreg = ahead[1 : 7 ,], n.ahead = 7 , lastn= FALSE )
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (720 ,738 ), main = "7 day MLR Forecast" , xlab = 'Time' , ylab = 'Total Users' )
points (seq (732 ,738 ,1 ),mlr_st_pred2$ pred,type = 'l' , pch = 15 ,col = 'blue' ,lwd= 2 , lty = 2 )
mlr_lt_pred2 = predict (mlr_forecast, newxreg = ahead[1 : 60 ,], n.ahead = 60 , lastn= FALSE )
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (700 ,785 ), main = "60 day MLR Forecast" , xlab = 'Time' , ylab = 'Total Users' )
points (seq (732 ,791 ,1 ),mlr_lt_pred2$ pred,type = 'l' , pch = 1 ,col = 'blue' ,lwd= 2 , lty = 2 )
# Year Out Forecast
mlr_lt_pred3 = predict (mlr_forecast, newxreg = ahead[1 : 365 ,], n.ahead = 365 , lastn= FALSE )
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (1 ,1100 ), main = "365 day MLR Forecast" , xlab = 'Time' , ylab = 'Total Users' )
points (seq (732 ,1096 ,1 ),mlr_lt_pred3$ pred,type = 'l' , pch = 1 ,col = 'blue' ,lwd= 1 , lty = 1 )
VARselect (daily_bike_data[,c (5 ,8 : 11 ,14 )])
$selection
AIC(n) HQ(n) SC(n) FPE(n)
7 7 7 7
$criteria
1 2 3 4 5
AIC(n) 2.762319e+01 2.733958e+01 2.716829e+01 2.689074e+01 2.632813e+01
HQ(n) 2.772619e+01 2.753087e+01 2.744788e+01 2.725862e+01 2.678429e+01
SC(n) 2.789002e+01 2.783512e+01 2.789256e+01 2.784372e+01 2.750982e+01
FPE(n) 9.922004e+11 7.472005e+11 6.296103e+11 4.770590e+11 2.718306e+11
6 7 8 9 10
AIC(n) -3.750199e+01 -4.271176e+01 -4.266486e+01 -4.261632e+01 -4.256334e+01
HQ(n) -3.695753e+01 -4.207901e+01 -4.194382e+01 -4.180699e+01 -4.166572e+01
SC(n) -3.609158e+01 -4.107264e+01 -4.079703e+01 -4.051977e+01 -4.023808e+01
FPE(n) 5.168071e-17 2.824148e-19 2.960966e-19 3.109863e-19 3.281202e-19
fit2 = VAR (daily_bike_data[,c (5 ,8 : 11 ,14 )], p = 7 , type = 'trend' )
plotts.wge (fit2$ varresult$ Total_Users$ residuals)
acf (fit2$ varresult$ Total_Users$ residuals)
ljung.wge (fit2$ varresult$ Total_Users$ residuals, p = 10 )
Obs 0.0005249251 -0.006235651 -0.007748364 -0.02024036 -0.01368047 -0.02223159 -0.02906443 -0.01706459 -0.04874905 0.01789249 -0.01377504 -0.01958757 -0.05185409 0.0353207 0.06875665 -0.01838377 0.01226764 -0.004110181 -0.01557603 0.005239665 0.05216921 0.007511638 0.001240346 0.01256811
$test
[1] "Ljung-Box test"
$K
[1] 24
$chi.square
[1] 13.29869
$df
[1] 14
$pval
[1] 0.5031507
ljung.wge (fit2$ varresult$ Total_Users$ residuals, p = 10 , K = 48 )
Obs 0.0005249251 -0.006235651 -0.007748364 -0.02024036 -0.01368047 -0.02223159 -0.02906443 -0.01706459 -0.04874905 0.01789249 -0.01377504 -0.01958757 -0.05185409 0.0353207 0.06875665 -0.01838377 0.01226764 -0.004110181 -0.01557603 0.005239665 0.05216921 0.007511638 0.001240346 0.01256811 -0.03178673 0.05470584 0.01272724 0.03989513 0.1070816 -0.04933038 0.01488104 -0.06148 0.02084819 0.0154614 3.051054e-05 -0.01613748 0.06611603 -0.05755259 0.04242315 -0.04065903 0.04235743 -0.001280656 -0.006977649 0.01617568 -0.037272 0.01445878 0.03332212 0.002087472
$test
[1] "Ljung-Box test"
$K
[1] 48
$chi.square
[1] 44.16066
$df
[1] 38
$pval
[1] 0.2274342
var_st_pred = predict (fit2, n.ahead = 7 , lastn = TRUE )
Warning in summary.lm(x): essentially perfect fit: summary may be unreliable
t = 1 : 731
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (670 ,731 ))
points (t[725 : 731 ], var_st_pred$ fcst$ Total_Users[,1 ], type= "l" , lwd= 2 , lty = 2 )
var_st_ase = mean ((daily_bike_data$ Total_Users[725 : 731 ]- var_st_pred$ fcst$ Total_Users[,1 ])^ 2 )
var_st_ase
var_lt_pred = predict (fit2, n.ahead = 60 , lastn = TRUE )
Warning in summary.lm(x): essentially perfect fit: summary may be unreliable
t = 1 : 731
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (672 ,731 ))
points (t[672 : 731 ], var_lt_pred$ fcst$ Total_Users[,1 ], type= "l" , lwd= 2 , lty = 1 )
var_lt_ase = mean ((daily_bike_data$ Total_Users[672 : 731 ]- var_lt_pred$ fcst$ Total_Users[,1 ])^ 2 )
var_lt_ase
var_st_pred2 = predict (fit2, n.ahead = 7 , lastn = FALSE )
Warning in summary.lm(x): essentially perfect fit: summary may be unreliable
t = 1 : 800
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (670 ,738 ), main = "Short Term VAR Forecast" )
points (t[732 : 738 ], var_st_pred2$ fcst$ Total_Users[,1 ], type= "l" , lwd= 2 , lty = 1 , col = 'blue' )
var_lt_pred2 = predict (fit2, n.ahead = 60 , lastn = FALSE )
Warning in summary.lm(x): essentially perfect fit: summary may be unreliable
t = 1 : 800
plot (seq (1 ,731 ,1 ),daily_bike_data$ Total_Users,type = 'l' ,xlim = c (670 ,791 ), main = "Long Term VAR Forecast" )
points (t[732 : 791 ], var_lt_pred2$ fcst$ Total_Users[,1 ], type= "l" , lwd= 2 , lty = 1 , col = 'blue' )